

where \(1 - \frac{\alpha }{c}\)  is the confidence level and \(\text{Beta}(\rho ;\varsigma , \vartheta )\)  is the \(\rho \) th quantile of the Beta distribution with shape parameters \(\varsigma \)  and \(\vartheta \) . Based on Bonferroni correction [1]}, [2]}, the overall confidence level for the \(c\)  label probability upper or lower bounds is \(1 - \alpha \) . To solve the optimization problem in Equation REF , we also need to estimate \(\underline{p_{A_u}}\)  and \(\overline{p}_{B_v}\) . In particular, we can estimate \(\underline{p_{A_u}} =\sum _{l=e^{\prime }}^{e^{\prime }+u-1}\underline{p_{a_l}}\)  and \(\overline{p}_{B_{v}} =\sum _{l=s-v+1}^{s}\overline{p}_{b_l}\) . However, this bound may be loose for \(\overline{p}_{B_{v}}\) . We can further improve the bound via considering the constraint that \( \overline{p}_{B_{v}} + \sum _{i \in L(\mathbf {x}) } \underline{p_i} \le k^{\prime } \) . In other words, we have \( \overline{p}_{B_{v}} \le k^{\prime }- \sum _{i \in L(\mathbf {x}) } \underline{p_i} \) . Given this constraint, we can estimate \(\overline{p}_{B_{v}} =\min (\sum _{l=s-v+1}^{s}\overline{p}_{b_l}, k^{\prime }- \sum _{i \in L(\mathbf {x}) } \underline{p_i})\) . Note that the above constraint is not applicable for \(\underline{p_{A_u}}\)  since it is a lower bound.
